Science China Mathematics

, Volume 61, Issue 8, pp 1421–1436 | Cite as

Functional inequalities on manifolds with non-convex boundary

  • Lijuan Cheng
  • Anton Thalmaier
  • James ThompsonEmail author


In this article, new curvature conditions are introduced to establish functional inequalities including gradient estimates, Harnack inequalities and transportation-cost inequalities on manifolds with non-convex boundary.


Ricci curvature gradient inequality log-Sobolev inequality geometric flow 


60J60 58J65 53C44 


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This work was supported by Fonds National de la Recherche Luxembourg (Grant No. O14/7628746 GEOMREV) and the University of Luxembourg (Grant No. IRP R-AGR-0517-10/AGSDE). The first author was supported by National Natural Science Foundation of China (Grant No. 11501508) and Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ16A010009).


  1. 1.
    Arnaudon M, Li X-M. Re ected Brownian motion: Selection, approximation and linearization. Electron J Probab, 2017, 22: 1–55CrossRefGoogle Scholar
  2. 2.
    Arnaudon M, Thalmaier A, Wang F-Y. Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below. Bull Sci Math, 2006, 130: 223–233MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Besse A L. Einstein Manifolds. Ergebnisse der Mathematik und Ihrer Grenzgebiete (3). Results in Mathematics and Related Areas (3), vol. 10. Berlin: Springer-Verlag, 1987Google Scholar
  4. 4.
    Bismut J-M. Large Deviations and the Malliavin Calculus. Progress in Mathematics, vol. 45. Boston: Birkhäuser, 1984Google Scholar
  5. 5.
    Elworthy K D, Li X-M. Formulae for the derivatives of heat semigroups. J Funct Anal, 1994, 125: 252–286MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Émery M. Stochastic Calculus in Manifolds. Berlin: Springer-Verlag, 1989CrossRefzbMATHGoogle Scholar
  7. 7.
    Hsu E P. Multiplicative functional for the heat equation on manifolds with boundary. Michigan Math J, 2002, 50: 351–367MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kasue A. A Laplacian comparison theorem and function theoretic properties of a complete Riemannian manifold. Jpn J Math, 1982, 8: 309–341MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Thalmaier A. On the diérentiation of heat semigroups and Poisson integrals. Stochastics Stochastic Rep, 1997, 61: 297–321CrossRefzbMATHGoogle Scholar
  10. 10.
    von Renesse M-K, Sturm K-T. Transport inequalities, gradient estimates, entropy, and Ricci curvature. Comm Pure Appl Math, 2005, 58: 923–940MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Wang F-Y. Functional Inequalities, Markov Semigroups and Spectral Theory. Beijing: Science Press, 2005Google Scholar
  12. 12.
    Wang F-Y. Gradient estimates and the first Neumann eigenvalue on manifolds with boundary. Stochastic Process Appl, 2005, 115: 1475–1486MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wang F-Y. Estimates of the first Neumann eigenvalue and the log-Sobolev constant on non-convex manifolds. Math Nachr, 2007, 280: 1431–1439MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Wang F-Y. Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds. Ann Probab, 2011, 39: 1449–1467MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wang F-Y. Transportation-cost inequalities on path space over manifolds with boundary. Doc Math, 2013, 18: 297–322MathSciNetzbMATHGoogle Scholar
  16. 16.
    Wang F-Y. Analysis for Diffusion Processes on Riemannian Manifolds. Advanced Series on Statistical Science Applied Probability, vol. 18. Hackensack: World Scientific, 2014Google Scholar
  17. 17.
    Wang F-Y. Modified curvatures on manifolds with boundary and applications. Potential Anal, 2014, 41: 699–714MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Lijuan Cheng
    • 1
    • 2
  • Anton Thalmaier
    • 1
  • James Thompson
    • 1
    Email author
  1. 1.Mathematics Research UnitUniversity of LuxembourgEsch-sur-AlzetteLuxembourg
  2. 2.Department of Applied MathematicsZhejiang University of TechnologyHangzhouChina

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